Higher order partial derivatives of a function of two variables

 Objective: Find all second order partial derivatives of a function of two variables

High order partial derivatives

Let's consider a function of two variables. We can calculate the partial derivatives of this function. If we calculate the partial derivatives of the first partial derivatives, we obtain second order partial derivatives. If we calculate the partial derivatives of these second order partial derivatives, we obtain third order partial  derivatives and so on.

Let's consider the function:

Its partial derivatives are:

The partial derivatives of these partial derivatives are:

𝛿²f/δx² = 8x and δ²f/δy² = -8x + 30y. These new partial derivatives are called second order partial derivatives.

There are 4 second order partial derivatives of any function provided they all exist

Example

Solution

Practice